The Astrophysical Journal, 723:1351–1358, 2010 November 10 doi:10.1088/0004-637X/723/2/1351C© 2010. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

MODELING MULTI-WAVELENGTH STELLAR ASTROMETRY. II. DETERMINING ABSOLUTEINCLINATIONS, GRAVITY-DARKENING COEFFICIENTS, AND SPOT PARAMETERS OF

SINGLE STARS WITH SIM LITE

Jeffrey L. Coughlin1,3

, Thomas E. Harrison1, and Dawn M. Gelino

21 Department of Astronomy, New Mexico State University, P.O. Box 30001, MSC 4500, Las Cruces, NM 88003-8001, USA; jlcough@nmsu.edu

2 NASA Exoplanet Science Institute, California Institute of Technology, Pasadena, CA 91125, USAReceived 2010 May 6; accepted 2010 September 8; published 2010 October 21

ABSTRACT

We present a novel technique to determine the absolute inclination of single stars using multi-wavelength sub-milliarcsecond astrometry. The technique exploits the effect of gravity darkening, which causes a wavelength-dependent astrometric displacement parallel to a star’s projected rotation axis. We find that this effect isclearly detectable using SIM Lite for various giant stars and rapid rotators, and present detailed models formultiple systems using the reflux code. We also explore the multi-wavelength astrometric reflex motion in-duced by spots on single stars. We find that it should be possible to determine spot size, relative tempera-ture, and some positional information for both giant and nearby main-sequence stars utilizing multi-wavelengthSIM Lite data. These data will be extremely useful in stellar and exoplanet astrophysics, as well as sup-porting the primary SIM Lite mission through proper multi-wavelength calibration of the giant star astromet-ric reference frame, and reduction of noise introduced by starspots when searching for extrasolar planets.

Key words: astrometry – stars: fundamental parameters

Online-only material: color figures

1. INTRODUCTION

SIM Lite is currently expected to have ∼80 spectral channels(Davidson et al. 2009), spanning 450–900 nm, thus allowingmulti-wavelength microarcsecond astrometry, which no currentor planned ground- or space-based astrometric project (GAIA,CHARA, VLT/PRIMA, etc.) is able to match. We showed inour first paper (Coughlin et al. 2010), hereafter referred toas Paper I, the implications multi-wavelength microarcsecondastrometry has for interacting binary systems. In this paper,we discuss an interesting effect we encountered while modelingbinary systems, namely, that gravity darkening in stars producesa wavelength-dependent astrometric offset from the center ofmass that increases with decreasing wavelength. It is possibleto use this effect to derive both the inclination and gravity-darkening exponent of a star in certain cases.

Determining the absolute inclination of a given star has manypractical applications. There is much interest in the formationof binary stars, where whether or not the spin axis of eachstar is aligned with the orbital axis provides insight into theformation history of the system (Turner et al. 1995). The mutualinclination between the stellar spin axes and the orbital axiscan greatly affect the rate of precession, which is used toprobe stellar structure and test general relativity (Sterne 1939a,1939b, 1939c; Kopal 1959; Jeffery 1984). Albrecht et al. (2009)recently reconciled a 30 year old discrepancy between theobserved and predicted precession rate of DI Herculis throughobservations which showed the stellar spin axes were nearlyperpendicular to the orbital axis. Along similar lines, extrasolarplanets discovered via the radial velocity technique only yieldthe planetary mass as a function of the inclination of theorbit (Mayor & Queloz 1995; Noyes et al. 1997; Marcy &Butler 2000), and thus, if one assumes the planetary orbit andstellar rotation axes are nearly parallel, determining the absolute

3 NSF Graduate Research Fellow.

inclination of the host star yields the absolute mass of the planet.If the stellar spin axis is found not to be parallel to the planetaryorbital axis, this provides valuable insights into the planet’sformation, migration, and tidal evolution histories (Winn et al.2006; Fabrycky & Winn 2009). A final example is the studyof whether or not the spin axes of stars in clusters are aligned,which both reveal insight into their formation processes as wellas significantly affect the determination of the distances to thoseclusters (Jackson & Jeffries 2010).

Our proposed technique can also be used in conjunction withother methods of determining stellar inclination to yield moreprecise inclination values and other stellar parameters of interest.Gizon & Solanki (2003) and Ballot et al. (2006) have shown thatone can derive the inclination of the rotation axis for a given starusing the techniques of asteroseismology given high-precisionphotometry with continuous coverage over a long baseline,such as that provided by the CoRoT and Kepler missions. Thistechnique is sensitive to rotation rates as slow as the Sun’s,but becomes easier with faster rotation rates. Domiciano deSouza et al. (2004) discuss how spectro-interferometry can yieldboth the inclination angle and amount of differential rotationfor a star, parameterized by α. For both eclipsing binaries andtransiting planets, the observation of the Rossiter–McLaughlin(RM) effect can yield the relative co-inclination between thetwo components (Winn et al. 2006; Albrecht et al. 2009;Fabrycky & Winn 2009). The technique we propose in this paperwould be complementary to these techniques in several ways.First, it would provide an independent check on the derivedinclination axis from each method, confirming or refutingthe asteroseismic models and spectro-interferometric and RMtechniques. Second, in principle the asteroseismic technique isnot dependent on the gravity-darkening coefficient β1, and thespectro-interferometric technique is correlated with the valuefor α; combining techniques would yield direct and robustobservationally determined values for i, α, and β1. Finally, theaccurate, observational determination of α and β1 (along with

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stellar limb darkening) is critical to accurately deriving the co-inclination from the RM effect, as well as other quantities instellar and exoplanet astrophysics.

In this paper, we also present models for multiple systemsand discuss the determination of spot location, temperature, andsize on single stars, which produce a wavelength-dependentastrometric signature as they rotate in and out of view. Starspotsare regions on the stellar surface where magnetic flux emergesfrom bipolar magnetic regions, which blocks convection andthus heat transport, effectively cooling the enclosed gas, and thusare fundamental indicators of stellar magnetic activity and theinternal dynamos that drive it. Isik et al. (2007) discuss how theobservation of spot location, duration, stability, and temperaturecan probe the stellar interior and constrain models of magneticflux transport. Through the observation of the rotation rates ofstarspots at varying latitudes, one is able to derive the differentialrotation rate of the star (Collier Cameron 2002), which may bedirectly related to the frequency of starspot cycles. Mappingspots in binary star systems provide insight into the interactionbetween the magnetic fields of the two components, which cancause orbital period changes (Applegate 1992), radii inflation(Lopez-Morales 2007; Morales et al. 2008), and may possiblyexplain the ∼2–3 hr period gap in cataclysmic variable systems(Watson et al. 2007). Detecting and characterizing starspots viamulti-wavelength astrometry would be complementary to otherexisting techniques, namely, optical interferometry (Wittkowskiet al. 2002), tomographic imaging (Donati et al. 2006; Auriereet al. 2008), photometric monitoring (Alekseev 2004; Mosseret al. 2009), and in the future, microlensing (Hwang & Han2010).

We present the details of our modeling code, reflux, inSection 2, discuss the inclination effect and present models formultiple stars in Section 3, discuss the spot effects and presentmodels in Section 4, and present our conclusions in Section 5.

2. THE reflux CODE

reflux4 is a code that computes the flux-weighted astromet-

ric reflex motion of binary systems. We discussed the code indetail in Paper I, but in short, it utilizes the Eclipsing LightCurve (ELC) code, which was written to compute light curvesof eclipsing binary systems (Orosz & Hauschildt 2000). TheELC code represents the surfaces of two stars as a grid of indi-vidual luminosity points, and calculates the resulting light curvegiven the provided systemic parameters. ELC includes the dom-inant physical effects that shape a binary’s light curve, such asnon-spherical geometry due to rotation, gravity darkening, limbdarkening, mutual heating, reflection effects, and the inclusionof hot or cool spots on the stellar surface. For the work in thispaper we have simply turned off one of the stars, thus allowingus to probe the astrometric effects of a single star. To computeintensity, ELC can either use a blackbody formula or interpolatefrom a large grid of NextGen model atmospheres (Hauschildtet al. 1999). For all the simulations in this paper, we have usedthe model atmosphere option, and will note now, and discuss inmore detail later, that the calculation of limb darkening is auto-matically included in NextGen model atmospheres. These arti-ficially derived limb-darkening coefficients have recently beenshown to be in error by as much as ∼10%–20% in comparisonto observationally derived values (Claret 2008), and thus their

4reflux can be run via a web interface from

http://astronomy.nmsu.edu/jlcough/reflux.html. Additional details as to how toset up a model are presented there.

uncertainties must be included, although for this work, due tosymmetry, we find the introduced error is negligible. For all oursimulations, we model the U, B, V, R, I, J, H, and K bands forcompleteness and comparison to future studies, though we notethat SIM Lite will not be able to observe in the U, J, H, or Kbandpasses.

3. INCLINATION AND ROTATION

The astrophysical phenomenon of gravity darkening, alsosometimes referred to as gravity brightening, is the drivingforce behind the ability to determine the inclination of asingle star using multi-wavelength astrometry. A rotating staris geometrically distorted into an oblate spheroid, such that itsequatorial radius is greater than its polar radius, and thus thepoles have a higher surface gravity, and the equator a lowersurface gravity, than a non-rotating star with the same massand average radius. This increased surface gravity, g, at thepoles results in a higher effective temperature, Teff , and thusluminosity; decreased g at the equator results in a lower Teff andluminosity. This temperature and luminosity differential causethe star’s center of light, or photocenter, to be shifted towardthe visible pole, away from the star’s gravitational center ofmass. Since the inclination determines how much of the poleis visible, the amount of displacement between the photocenterand the center of mass is directly related to the inclination.Furthermore, since the luminosity difference effectively resultsfrom a ratio of blackbody luminosities of differing temperatures,the effect is wavelength dependent, with shorter wavelengthsshifted more than longer wavelengths. Thus, the amount ofdisplacement between the measured photocenter in two or morewavelengths is directly related to the inclination. See Figure 1for an illustration of the effect.

An additional complicating factor is the exact dependence oftemperature on local gravity. von Zeipel (1924) was the firstto derive the quantitative relationship between them, showingthat T 4

eff ∝ gβ1 , where β1 is referred to as the gravity-darkeningexponent. The value of β1 has been a subject of much study anddebate; for a complete review, see Claret (2000), who presentsboth an excellent discussion of past studies, as well as new,detailed computations of β1 using modern models of stellaratmospheres and internal structure that encompass stars from0.08 to 40 M�. Since the value of β1 affects the temperaturedifferential between equator and pole, the multi-wavelengthdisplacement will also be dependent on the value of β1. Thetotal amplitude of the effect will be scaled by the angular size ofthe star, which depends on both its effective radius and distance.Thus, in total, the components of this inclination effect are theeffective stellar radius, distance, effective temperature, rotationrate, β1, and inclination of the star. In principle, one is ableto determine the effective stellar radius, effective temperature,rotation rate, and distance of a target star using ground-based spectroscopy and space-based parallax measurements,including from SIM Lite. Thus, when modeling the multi-wavelength displacement of the stellar photocenter, the onlytwo components that need to be solved for are the inclinationand β1, with β1 already having some constraints from theory.

A good trio of stars for modeling and testing this inclinationeffect is the components of the binary system Capella (Aa andAb) and the single star Vega. Torres et al. (2009) have veryrecently published an extremely detailed analysis of both thebinary orbit of Capella and the physical and evolutionary statesof the individual components, providing both new observations,as well as drawing from the previous observations and analyses

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Figure 1. Illustration of the inclination effect using brightness maps of Capella Ab, which has an inclination of 42.◦788 (Torres et al. 2009). Left: Capella Ab artificiallyspun-up to near its break-up speed to accentuate the gravity-darkening effect for ease of viewing. As can be seen, the photocenter of the system is dramatically shiftedaway from the center of mass, toward the visible pole, which is brighter than the rest of the star due to gravity darkening. Furthermore, since the pole is physicallyhotter, the U-band photocenter is shifted more than the K-band photocenter, and in the direction of the projected rotation axis or the y-axis. Right: Capella Ab at itsactual rotation period. As can be seen, the actual effect is small compared to the angular size of the star on the sky, but still large compared to the 1 μas benchmark ofSIM Lite. The presence of limb darkening is clearly visible, which causes a decrement in flux toward the limb of the star. Note that the broad wavelength coverage ofSIM Lite will only cover the B, V, R, and I bandpasses.

(A color version of this figure is available in the online journal.)

Table 1Parameters for Capella Aa

Parameter Valuea

Distance (pc) 12.9Rotation period (days) 106.0Mass (M�) 2.70Radius (R�) 12.2Effective temperature (K) 4940β1 0.43

Note. a Values from Torres et al. (2009).

of Hummel et al. (1994) and Strassmeier et al. (2001). Vega,in addition to being one of the most well-studied stars in thesky, has recently been discovered to be a very rapid rotator seennearly pole-on (Aufdenberg et al. 2006; Peterson et al. 2006;Hill et al. 2010). In total, these three stars represent both slowand rapid rotators for giant and main-sequence stars at a rangeof temperatures, as Capella Aa is a slow-rotating K-type giant,Capella Ab is a fast-rotating G-type giant, and Vega is a very fastrotating A-type main-sequence star. With many ground-basedinterferometric observations to compare with, and being brightand nearby, these stars also present excellent targets for SIMLite.

We use the reflux code to generate models of the astrometricdisplacement from U band to H band, with respect to theK-band photocenter, for inclinations from 0◦ to 90◦, for eachstar, as shown in Figures 2–4. We use systemic parametersgiven by Torres et al. (2009) for Capella Aa and Ab, andby Aufdenberg et al. (2006) and Peterson et al. (2006) forVega, listed in Tables 1, 2, and 3, respectively. We employthe model atmospheres incorporated into the ELC code, as wellas automatically chosen values for β1 based on Figure 1 ofClaret (2000). Additionally, in each figure we show a dashed

Table 2Parameters for Capella Ab

Parameter Valuea

Distance (pc) 12.9Rotation period (days) 8.64Mass (M�) 2.56Radius (R�) 9.2Effective temperature (K) 5700β1 0.39

Note. a Values from Torres et al. (2009).

Table 3Parameters for Vega

Parameter Valuea

Distance (pc) 7.76Rotation period (days) 0.521Mass (M�) 2.11Radius (R�) 2.5Effective temperature (K) 9602β1 1.02

Note.a Values from Aufdenberg et al. (2006) and Peterson et al.(2006).

line to indicate the effect of decreasing the gravity-darkeningcoefficient by 10% to simulate the uncertainty of the models(Claret 2000) and explore the correlation with other parameters.

As can be seen from these models, we find that the effectis quite large for a Capella Ab-like or Vega-like fast rotator,but only marginally detectable for a slower rotating system likeCapella Aa. This also implies that this effect would not bedetectable for a slow-rotating, main-sequence star like our Sun.

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Figure 2. Astrometric displacement of each bandpass with respect to K band vs. inclination for Capella Aa. Dashed lines are a model with β1 decreased by 10%. Dueto the slow rotation rate of Capella Aa, the effect is limited to a maximum of ∼2.0 μas between U and K bands, and only a maximum of ∼0.7 μas between B and Ibands, where SIM Lite will operate. This puts the detection of this effect for Capella Aa at the very edge of SIM Lite’s capability.

(A color version of this figure is available in the online journal.)

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Figure 3. Astrometric displacement of each bandpass with respect to K band vs. inclination for Capella Ab. Dashed lines are a model with β1 decreased by 10%.Due to the fast rotation rate of a Capella Ab like star, the effect is moderate with tens of microarcseconds of displacement, and thus these types of stars are excellenttargets for SIM Lite. The actual inclination of Capella Ab is 42.◦788 (Torres et al. 2009), and thus Capella Ab itself should show a large shift of the photocenter withwavelength. Note that the broad wavelength coverage of SIM Lite will only cover the B, V, R, and I bandpasses.

(A color version of this figure is available in the online journal.)

Our modeling confirms this, showing a total U–K amplitude of�0.1 μas for a 1.0 M�, 1.0 R� star with a rotation period of30.0 days at 10.0 parsec. These conclusions on detectability are

made with the assumption that, for bright stars like these, SIMLite can achieve its microarcsecond benchmark. We show thatthis is possible in a narrow angle (NA) mode by employing the

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Figure 4. Astrometric displacement of each bandpass with respect to K band vs. inclination for Vega. Dashed lines are a model with β1 decreased by 10%. Note thatdue to the very fast rotation of Vega, along with a high value of β1, the effect can be quite large, at a couple of hundred microarcseconds. For a Vega-like star, SIM Liteobservations would yield very accurate values for β1 and the inclination. For Vega itself, which is known to be nearly pole on, with an inclination of 5.◦7 (Hill et al.2010), there should be a B-band minus I-band displacement of 6.0 μas, still detectable by SIM Lite. Note that the broad wavelength coverage of SIM Lite will onlycover the B, V, R, and I bandpasses.

(A color version of this figure is available in the online journal.)

SIM Differential Astrometry Performance Estimator (DAPE;Plummer 2009). For a target star with magnitude V = 5, and asingle comparison star with V = 10 located within a degree ofit on the sky, by integrating 15 s on the target, and 30 s on thereference, for 10 visits at five chop cycles each, a final precisionof ±1.01 μas is achieved in only 1.04 hr of the total mission time.For a fainter target with V = 10, this precision is only reducedto ±1.32 μas in the same amount of mission time. In utilizingNA mode, one must be careful in choosing the reference star(s)to ensure that they are not stars with a substantial wavelength-dependent centroid. Given the only constraints on referencestars are that they need to have V � 10 and are within 1◦ onthe sky, one could easily choose a slow-rotating, main-sequencestar, determined as such via ground-based observations as awavelength-independent astrometric reference star. We also notethat wide angle SIM Lite measurements, with a precision of∼5 μas, may not detect the wavelength-dependent photocenterof a system like Capella, but will have no difficulty detecting itin stars like Capella Ab or Vega.

The effect of decreasing the gravity-darkening exponent isto decrease the total amplitude of the effect in each wave-length, with shorter wavelengths affected more than longerwavelengths. Thus, the choice of gravity-darkening exponent isintimately tied to the derived inclination. If one were to modelobserved data with a gravity-darkening exponent that was ∼10%different than the true value, one would derive an inclination thatwould also be ∼10% different from the true inclination. How-ever, the two combinations of inclination and gravity-darkeningexponent do not produce identical results, and can be distin-guished with a sufficient precision at a number of wavelengths.For example, if one were to adopt the nominal value for β1 andderive an inclination of 40◦ for a Vega-like star, then adopt a

β1 value that was 10% lower, one would derive an inclinationof 43◦, a 7.5% change. In this case though, with the lower β1value, the measured photocenter in the U, B, V, R, I, J, andH bandpasses, with respect to the K-band photocenter, woulddiffer from the nominal β1 model by ∼0.5, −1.0, −2.0, −2.0,−1.6, −1.0, and 0.2 μas, respectively. Note that for B, V, R,and I, where SIM Lite can observe, these discrepancies, on theorder of ∼1.0 μas, should be large enough to be distinguishedin the NA mode. Thus, a unique solution exists for the values ofi and β1 if the photocenter is measured in three or more wave-lengths. (The photocenter of one wavelength is used as a basemeasurement, with respect to which the photocenters of otherwavelengths are measured. If one measures the photocenter inthree or more wavelengths, one then has two or more differen-tial photocenter measurements, and can then solve for the twounknown variables. Although we have chosen the K band asthe base measurement in our models, simply because it is thelongest wavelength we model and thus shows the least variation,one could easily choose another band, such as R, where SIM Litewill be able to observe. In the example just given of possibledegeneracy between inclination and β1, if one chose the R bandas the base measurement, one would still detect a difference of∼1.0 μas between the B and V photocenters, and be able toresolve the degeneracy between inclination and β1.)

Another complication is the possibility of having equallygood fitting high and low solutions for i. For example, if oneobserved and determined a best-fit inclination of 70◦ for a Vega-like star, one could obtain a reasonably good fit as well at 46◦,(see Figure 4). However, just as in the case of the uncertaintyin the value of β1, discernible discrepancies would exist. In thiscase, the discrepancies in the measured photocenter in the U,B, V, R, I, J, and H bandpasses, with respect to the K-band

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photocenter, would be ∼0.1, −9.0, −2.0, 1.5, 6.0, 1.0, and0.2 μas, respectively. Just as in the case of the uncertainty in thevalue of β1, this discrepancy between equally good fitting high-and low-inclination solutions can be resolved if one has three ormore wavelengths obtained in the NA mode.

As mentioned in Section 1, we note that the limb-darkeningfunction, which was automatically chosen by the ELC code asincorporated into the model atmospheres, can differ from actualobserved values by ∼10% (Claret 2008). We have tested howchanging the limb-darkening coefficients by 10% affects theresulting astrometric displacements, and find that the result isless than 0.5% for all wavelengths, and thus is negligible in themodeling. The reason is that limb darkening is symmetric, andthus while increased limb darkening damps the visible pole, italso damps the rest of the star, and thus the relative brightnessbetween regions is maintained.

Additionally, this inclination technique yields the orientationof the projected stellar rotation axis on the sky, which is parallelto the wavelength dispersion direction. When coupled with thederived inclination, this technique thus yields the full three-dimensional orientation of the rotation axis. This could be apowerful tool in determining the overall alignment of stellaraxes in the local neighborhood and in nearby clusters.

4. STARSPOTS

Another area of astrophysical interest to which multi-wavelength astrometric measurements from SIM Lite can con-tribute is the study of starspots. As the cause of starspots areintense magnetic fields at the photosphere, they are typicallyfound in stars with convective envelopes, especially rapidlyrotating stars. Thus, both low-mass, main-sequence K and Mdwarfs, as well as rapidly rotating giant and sub-giant stars, areknown to host large spots on their surface. The study of thedistribution, relative temperature, and size of these spots wouldgreatly contribute to the study of magnetic field generation instellar envelopes. A starspot that rotates in and out of viewwill cause a shift of the photocenter for a single star, whichhas been a subject of much recent discussion in the literature(e.g., Hatzes 2002; Unwin 2005; Eriksson & Lindegren 2007;Catanzarite et al. 2008; Makarov et al. 2009; Lanza et al. 2008),especially in light of its potential to mimic, or introduce noisewhen characterizing, an extrasolar planet. However, there hasbeen no mention in the literature of the multi-wavelength astro-metric signature of stellar spots, where, just as in the case ofthe gravity-darkening inclination effect, we are looking at es-sentially two blackbodies with varying temperatures, and thusshorter wavelengths will be more affected by a spot than longerwavelengths.

To characterize the multi-wavelength astrometric signatureof stellar spots, we model two spotty systems, again usingthe reflux code. We model Capella Ab, which shows evidenceof large spots and is suspected of being an RS CVn variable(Hummel et al. 1994), and a typical main-sequence K dwarf.For Capella Ab, we use the parameters listed in Table 2, alongwith the star’s determined inclination of 42.◦788 (Torres et al.2009), and add a cool spot that has a temperature that is 60%of the average surface temperature, located at the equator, at alongitude such that it is seen directly at phase 270◦, and havingan angular size of 10◦ (where 90◦ would cover exactly one half ofthe star). For the K dwarf system, we use the physical parameterslisted in Table 4, simulating a typical K Dwarf at 10 pc, and adda cool spot with the same parameters as we do for Capella Ab.Additionally, to investigate the effects of cool versus hot spots

Table 4Parameters for the K Dwarf System

Parameter Value

Distance (pc) 10.0Inclination (◦) 60.0Period (days) 20Mass (M�) 0.6Radius (R�) 0.6Effective temperature (K) 4500Latitude of spot (◦) 90Longitude of spot (◦) 270Angular Size of Spot (◦) 10Cool spot temperature factor 0.6Hot spot temperature factor 1.4

or flares, we also run a model with a hot spot by changing thespot temperature to be 40% greater than the average surfacetemperature. We present our models in Figures 5–7.

As can be seen for Capella Ab, the gravity-darkening inclina-tion effect presented in Section 3 dominates the spread of colorsin the y-direction, the direction parallel to the stars’ projectedrotation axis. However, the amplitude of the spot motion is quitelarge, with a total amplitude of ∼40 μas in all bandpasses, whichwould be easily detectable by SIM Lite. For the K dwarf witha cool spot, we see a much smaller, but still detectable shiftof amplitude ∼5–8 μas, depending on the wavelength. In thecase of a hot spot or flare, we see a much larger displacement,on the order of ∼10–200 μas, depending on the wavelength,which would be easily detectable by SIM and provide extremelyprecise values in deriving the spot parameters.

In general, the temperature of the spot, in relation to the meanstellar surface temperature, is related to the spread in observedwavelengths, with a larger spread indicating a larger temperaturedifference. The duration of the astrometric displacement inphase, coupled with the overall amplitude of the astrometricdisplacement, yields the size of the spot, as larger spots willcause larger displacements and be visible for a larger amount ofrotational phase. The latitude of the spot can also affect the totalduration. Finally, the amplitude of the astrometric displacementin the x direction versus the y direction is dependent on boththe latitude of spot and the inclination of the star. Thus, whenmodeled together, one is able to recover these parameters. Thiswork can also be combined with our work in Paper I to derive thelocation of spots in binary systems, as the astrometric signatureof the spot is simply added to the astrometric signature of thebinary system.

The astrometric motion induced upon a parent star by a hostplanet does not have a wavelength dependence. Spots however,as we have shown via our modeling, have a clear wavelengthdependence. Thus, if one has a candidate planetary signal fromastrometry, but it shows a wavelength-dependent motion, it mustthen be a false positive introduced from starspots at the rotationperiod of the star (assuming that the planet’s emitted flux isnegligible compared to the star). Furthermore, when SIM islaunched, there will likely be many cases where a marginallydetectable signal due to a planetary companion is found ata very different period than the rotation period of the star.However, starspots will still introduce extra astrometric jitterwhich will degrade the signal from the planetary companion.Multi-wavelength astrometric data can be used to model andremove the spots, which will have a wavelength dependence,and thus strengthen the planetary signal, which will not have awavelength dependence.

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Figure 6. Simulated cool spot on a nearby K dwarf star with an inclination of 60◦, whose parameters are given in Table 4. The spot is located on the equator, with alongitude such that it is seen directly at phase 270◦. Note that the broad wavelength coverage of SIM Lite will only cover the B, V, R, and I bandpasses.

(A color version of this figure is available in the online journal.)

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Figure 7. Simulated hot spot or flare on a nearby K dwarf star with an inclination of 60◦, whose parameters are given in Table 4. The spot is located on the equator,with a longitude such that it is seen directly at phase 270◦. Note that the broad wavelength coverage of SIM Lite will only cover the B, V, R, and I bandpasses.

(A color version of this figure is available in the online journal.)

1358 COUGHLIN, HARRISON, & GELINO Vol. 723

5. DISCUSSION AND CONCLUSION

We have presented detailed models of the multi-wavelengthastrometric displacement that SIM Lite will observe due togravity darkening and stellar spots using the reflux code.We find that SIM Lite observations, especially when combinedwith other techniques, will be able to determine the absoluteinclination, gravity-darkening exponent, and three-dimensionalorientation of the rotational axis for fast- and slow-rotatinggiant stars and fast-rotating main-sequence stars. This techniquewill be especially useful in probing binary star and exoplanetformation and evolution, as well as the physics of star-formingregions. Direct observational determination of the gravity-darkening exponent has direct applications in both stellarand exoplanet astrophysics. This technique is also relativelyinexpensive in terms of SIM Lite observing time, as one needonly to observe a given star once, as opposed to binary stars andplanets, which require constant monitoring over an entire orbit.It should be noted that this effect should be taken into accountwhen constructing the SIM Lite astrometric reference frame,such that fast-rotating giants should be excluded so as not toproduce a wavelength-dependent astrometric reference frame.

We have also presented models of starspots on single starsand find that SIM Lite should be able to discern their location,temperature, and size. Combined with other techniques, this willprovide great insight into stellar differential rotation, magneticcycles and underlying dynamos, and magnetic interaction inclose binaries. From this modeling, it should especially be notedthat multi-wavelength astrometry is a key tool in the hunt forextrasolar planets, either by ruling out false signals created byspots or simply removing extra astrometric jitter introduced byspots. Thus, it remains critical that SIM Lite maintains a multi-wavelength astrometric capability in its final design.

This work was sponsored in part by a SIM Science Study (PI:D. Gelino) from the National Aeronautics and Space Adminis-tration through a contract with the Jet Propulsion Laboratory,California Institute of Technology. J.L.C. acknowledges addi-tional support from a New Mexico Space Grant ConsortiumFellowship. We thank the referee for comments which greatlyhelped to improve this manuscript, particularly in making thepresentation of our ideas much clearer.

REFERENCES

Albrecht, S., Reffert, S., Snellen, I. A. G., & Winn, J. N. 2009, Nature, 461, 373Alekseev, I. Y. 2004, Sol. Phys., 224, 187Applegate, J. H. 1992, ApJ, 385, 621

Aufdenberg, J. P., et al. 2006, ApJ, 645, 664Auriere, M., et al. 2008, A&A, 491, 499Ballot, J., Garcıa, R. A., & Lambert, P. 2006, MNRAS, 369, 1281Catanzarite, J., Law, N., & Shao, M. 2008, Proc. SPIE, 7013, 70132KClaret, A. 2000, A&A, 359, 289Claret, A. 2008, A&A, 482, 259Collier Cameron, A. 2002, Astron. Nachr., 323, 336Coughlin, J. L., et al. 2010, ApJ, 717, 776Davidson, J., Edberg, S., Danner, R., Nemati, B., & Unwin, S. (ed.) 2009, SIM

Lite: Astrometric Observatory (Pasadena, CA: NASA JPL)Domiciano de Souza, A., Zorec, J., Jankov, S., Vakili, F., Abe, L., & Janot-

Pacheco, E. 2004, A&A, 418, 781Donati, J., Forveille, T., Cameron, A. C., Barnes, J. R., Delfosse, X., Jardine,

M. M., & Valenti, J. A. 2006, Science, 311, 633Eriksson, U., & Lindegren, L. 2007, A&A, 476, 1389Fabrycky, D. C., & Winn, J. N. 2009, ApJ, 696, 1230Gizon, L., & Solanki, S. K. 2003, ApJ, 589, 1009Hatzes, A. P. 2002, Astron. Nachr., 323, 392Hauschildt, P. H., Allard, F., & Baron, E. 1999, ApJ, 512, 377Hill, G., Gulliver, A. F., & Adelman, S. J. 2010, ApJ, 712, 250Hummel, C. A., Armstrong, J. T., Quirrenbach, A., Buscher, D. F., Mozurkewich,

D., Elias, II, N. M., & Wilson, R. E. 1994, AJ, 107, 1859Hwang, K., & Han, C. 2010, ApJ, 709, 327Isik, E., Schussler, M., & Solanki, S. K. 2007, A&A, 464, 1049Jackson, R. J., & Jeffries, R. D. 2010, MNRAS, 402, 1380Jeffery, C. S. 1984, MNRAS, 207, 323Kopal, Z. 1959, in Close Binary Systems, ed. Z. Kopal (London: Chapman &

Hall)Lanza, A. F., De Martino, C., & Rodono, M. 2008, New Astron., 13, 77Lopez-Morales, M. 2007, ApJ, 660, 732Makarov, V. V., Beichman, C. A., Catanzarite, J. H., Fischer, D. A., Lebreton,

J., Malbet, F., & Shao, M. 2009, ApJ, 707, L73Marcy, G. W., & Butler, R. P. 2000, PASP, 112, 137Mayor, M., & Queloz, D. 1995, Nature, 378, 355Morales, J. C., Ribas, I., & Jordi, C. 2008, A&A, 478, 507Mosser, B., Baudin, F., Lanza, A. F., Hulot, J. C., Catala, C., Baglin, A., &

Auvergne, M. 2009, A&A, 506, 245Noyes, R. W., Jha, S., Korzennik, S. G., Krockenberger, M., Nisenson, P., Brown,

T. M., Kennelly, E. J., & Horner, S. D. 1997, ApJ, 483, L111Orosz, J. A., & Hauschildt, P. H. 2000, A&A, 364, 265Peterson, D. M., et al. 2006, Nature, 440, 896Plummer, K. 2009, TaPE Webtool User Guide 1.1Sterne, T. E. 1939a, MNRAS, 99, 451Sterne, T. E. 1939b, MNRAS, 99, 662Sterne, T. E. 1939c, MNRAS, 99, 670Strassmeier, K. G., Reegen, P., & Granzer, T. 2001, Astron. Nachr., 322, 115Torres, G., Claret, A., & Young, P. A. 2009, ApJ, 700, 1349Turner, J. A., Chapman, S. J., Bhattal, A. S., Disney, M. J., Pongracic, H., &

Whitworth, A. P. 1995, MNRAS, 277, 705Unwin, S. C. 2005, in ASP Conf. Ser. 338, Astrometry in the Age of the Next

Generation of Large Telescopes, ed. P. K. Seidelmann & A. K. B. Monet(San Francisco, CA: ASP), 37

von Zeipel, H. 1924, MNRAS, 84, 665Watson, C. A., Steeghs, D., Shahbaz, T., & Dhillon, V. S. 2007, MNRAS, 382,

1105Winn, J. N., et al. 2006, ApJ, 653, L69Wittkowski, M., Scholler, M., Hubrig, S., Posselt, B., & von der Luhe, O.

2002, Astron. Nachr., 323, 241